Levy's Lemma asserts Lipschitz functions of vectors chosen uniformly from the unit hypersphere concentrate:

Lemma. Suppose $f:\mathbb{S}^{d-1} \to \mathbb{R}$ is $L$-Lipschitz on the unit hypersphere. Then, if $\vec{x}$ is drawn uniformly from the $d$-dimensional unit hypersphere, for some constant $C>0$, \begin{equation*} \mathbb{P}[ |f(\vec{x}) - \mathbb{E}[f(\vec{x})]| > \epsilon ] \leq 2\exp\left( \frac{-C(d+1)\epsilon^2}{L^2} \right). \end{equation*}

I am interested in 2-norm bounds for the vector case ($f:\mathbb{S}^{d-1}\to\mathbb{R}^n$ and $f$ satisfies $\|f(\vec{x}) - f(\vec{y})\|_2 \leq L \| \vec{x}-\vec{y}\|_2$). In particular, I am wondering whether an $n$ dependence is necessary? I have tried searching for such a result as I imagine this problem is standard, but haven't seemed to find the right keyword.

Levy's Lemma asserts Lipschitz functions of vectors chosen uniformly from the unit hypersphere concentrate:

Lemma. Suppose $f:\mathbb{S}^{d-1} \to \mathbb{R}$ is $L$-Lipschitz on the unit hypersphere. Then, if $\vec{x}$ is drawn uniformly from the $d$-dimensional unit hypersphere, for some constant $C>0$, \begin{equation*} \mathbb{P}[ |f(\vec{x}) - \mathbb{E}[f(\vec{x})]| > \epsilon ] \leq 2\exp\left( \frac{-C(d+1)\epsilon^2}{L^2} \right). \end{equation*}

I am interested in 2-norm bounds for the vector case ($f:\mathbb{S}^{d-1}\to\mathbb{R}^n$ and $f$ satisfies $\|f(\vec{x}) - f(\vec{y})\|_2 \leq L \| \vec{x}-\vec{y}\|_2$). In particular, I am wondering whether a $d$ dependence is necessary? I have tried searching for such a result as I imagine this problem is standard, but haven't seemed to find the right keyword.

Levy's Lemma asserts Lipshitz functions of vectors chosen uniformly from the unit hypersphere concentrate:

Lemma. Suppose $f:\mathbb{S}^{d-1} \to \mathbb{R}$ is $L$-Lipshitz on the unit hypersphere. Then, if $\vec{x}$ is drawn uniformly from the $d$-dimensional unit hypersphere, for some constant $C>0$, \begin{equation*} \mathbb{P}[ |f(\vec{x}) - \mathbb{E}[f(\vec{x})]| > \epsilon ] \leq 2\exp\left( \frac{-C(d+1)\epsilon^2}{L^2} \right). \end{equation*}

I am interested in 2-norm bounds for the vector case ($f:\mathbb{S}^{d-1}\to\mathbb{R}^n$ and $f$ satisfies $\|f(\vec{x}) - f(\vec{y})\|_2 \leq L \| \vec{x}-\vec{y}\|_2$). In particular, I am wondering whether an $n$ dependence is necessary? I have tried searching for such a result as I imagine this problem is standard, but haven't seemed to find the right keyword.

Levy's Lemma asserts Lipschitz functions of vectors chosen uniformly from the unit hypersphere concentrate:

Lemma. Suppose $f:\mathbb{S}^{d-1} \to \mathbb{R}$ is $L$-Lipschitz on the unit hypersphere. Then, if $\vec{x}$ is drawn uniformly from the $d$-dimensional unit hypersphere, for some constant $C>0$, \begin{equation*} \mathbb{P}[ |f(\vec{x}) - \mathbb{E}[f(\vec{x})]| > \epsilon ] \leq 2\exp\left( \frac{-C(d+1)\epsilon^2}{L^2} \right). \end{equation*}

I am interested in 2-norm bounds for the vector case ($f:\mathbb{S}^{d-1}\to\mathbb{R}^n$ and $f$ satisfies $\|f(\vec{x}) - f(\vec{y})\|_2 \leq L \| \vec{x}-\vec{y}\|_2$). In particular, I am wondering whether an $n$ dependence is necessary? I have tried searching for such a result as I imagine this problem is standard, but haven't seemed to find the right keyword.